Simplify the following expression: $y = \dfrac{-8x^2+31x+45}{x - 5}$
Solution: First use factoring by grouping to factor the expression in the numerator. This expression is in the form ${A}x^2 + {B}x + {C}$ First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(-8)}{(45)} &=& -360 \\ {a} + {b} &=& &=& {31} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $-360$ and add them together. Remember, since $-360$ is negative, one of the factors must be negative. The factors that add up to ${31}$ will be your ${a}$ and ${b}$ When ${a}$ is ${-9}$ and ${b}$ is ${40}$ $ \begin{eqnarray} {ab} &=& ({-9})({40}) &=& -360 \\ {a} + {b} &=& {-9} + {40} &=& 31 \end{eqnarray} $ Next, rewrite the expression as $({A}x^2 + {a}x) + ({b}x + {C})$ $ ({-8}x^2 {-9}x) + ({40}x +{45}) $ Factor out the common factors: $ x(-8x - 9) - 5(-8x - 9)$ Now factor out $(-8x - 9)$ $ (-8x - 9)(x - 5)$ The original expression can therefore be written: $ \dfrac{(-8x - 9)(x - 5)}{x - 5}$ We are dividing by $x - 5$ , so $x - 5 \neq 0$ Therefore, $x \neq 5$ This leaves us with $-8x - 9; x \neq 5$.